We present a new solution framework to solve the generalized trust region subproblem (GTRS) of minimizing a quadratic objective over a quadratic constraint. More specifically, we derive a convex quadratic reformulation (CQR) via minimizing a linear objective over two convex quadratic constraints for the GTRS. We show that an optimal solution of the GTRS can be recovered from an optimal solution of the CQR. We further prove that this CQR is equivalent to minimizing the maximum of the two convex quadratic functions derived from the CQR for the case under our investigation. Although the latter minimax problem is nonsmooth, it is well-structured and convex. We thus develop two steepest descent algorithms corresponding to two different line search rules. We prove for both algorithms their global sublinear convergence rates. We also obtain a local linear convergence rate of the first algorithm by estimating the Kurdyka-Lojasiewicz exponent at any optimal solution under mild conditions. We finally demonstrate the efficiency of our algorithms in our numerical experiments. Problem (P) is known as the generalized trust region subproblem (GTRS) [44,41]. When Q 2 is an identity matrix I and b 2 = 0, c = −1/2, problem (P) reduces to the classical trust region subproblem (TRS). The TRS first arose in the trust region method for nonlinear optimization [15,49], and has found many applications including robust optimization [8] and the least square problems [50]. As a generalization, the GTRS also admits its own applications such as time of arrival problems [26] and subproblems of consensus ADMM in signal processing [29]. Over the past two decades, numerous solution methods have been developed for TRS (see [38,36,48,42,25,22,4] and references therein).Various methods have been developed for solving the GTRS under various assumptions (see [37,44,10,45,16,41,5] and references therein). Although it appears being nonconvex, the GTRS essentially enjoys Assumption 2.1. The set I P SD := {λ : Q 1 + λQ 2 0} ∩ R + is not empty, where R + is the nonnegative orthant.Assumption 2.2. The common null space of Q 1 and Q 2 is trivial, i.e., Null(Q 1 ) ∩ Null(Q 2 ) = {0}.Before introducing our CQR, let us first recall the celebrated S-lemma by definingf 1 (x) = f 1 (x) + γ with an arbitrary constant γ ∈ R.Lemma 2.3 (S-lemma [47,40]). The following two statements are equivalent: 1. The system off 1 (x) < 0 and f 2 (x) ≤ 0 is not solvable; 2. There exists µ ≥ 0 such thatf 1 (x) + µf 2 (x) ≥ 0 for all x ∈ R n .Using the S-lemma, the following lemma shows a necessary and sufficient condition under which problem (P) is bounded from below.

We investigate in this paper the generalized trust region subproblem (GTRS) of minimizing a general quadratic objective function subject to a general quadratic inequality constraint. By applying a simultaneous block diagonalization approach, we obtain a congruent canonical form for the symmetric matrices in both the objective and constraint functions. By exploiting the block separability of the canonical form, we show that all GTRSs with an optimal value bounded from below are second order cone programming (SOCP) representable. Our result generalizes the recent work of Ben-Tal and Hertog (Math. Program. 143(1-2):1-29, 2014), which establishes the SOCP representability of the GTRS under the assumption of the simultaneous diagonalizability of the two matrices in the objective and constraint functions. Compared with the state-of-the-art approach to reformulate the GTRS as a semi-definite programming problem, our SOCP reformulation delivers a much faster solution algorithm. We further extend our method to two variants of the GTRS in which the inequality constraint is replaced by either an equality constraint or an interval constraint. Our methods also enable us to obtain simplified versions of the classical S-lemma, the S-lemma with equality, and the S-lemma with interval bounds.

Abstract. An equivalence between attainability of simultaneous diagonalization (SD) and hidden convexity in quadratically constrained quadratic programming (QCQP) stimulates us to investigate necessary and sufficient SD conditions, which is one of the open problems posted by HiriartUrruty [SIAM Rev., 49 (2007), pp. 255-273] nine years ago. In this paper we give a necessary and sufficient SD condition for any two real symmetric matrices and offer a necessary and sufficient SD condition for any finite collection of real symmetric matrices under the existence assumption of a semi-definite matrix pencil. Moreover, we apply our SD conditions to QCQP, especially with one or two quadratic constraints, to verify the exactness of its second-order cone programming relaxation and to facilitate the solution process of QCQP.Key words. simultaneous diagonalization, congruence, quadratically constrained quadratic programming, second-order cone programming relaxation.AMS subject classifications. 15A, 65K, 90C1. Introduction. Recently, Ben-Tal and Hertog [2] show that if the two matrices in the objective function and the constraint of a quadratically constrained quadratic programming (QCQP) problem are simultaneously diagonalizable (SD), this QCQP problem can be then turned into an equivalent second-order cone programming (SOCP) problem, which can be solved much faster than the semi-definite programming (SDP) reformulation. (Note that to simply the notation, we use the abbreviation SD to denote both "simultaneously diagonalizable" and "simultaneous diagonalization" via congruence when no confusion will arise.) A natural question to ask is when a given pair of matrices are SD. Essentially, the SD problem of a finite collection of symmetric matrices via congruence is one of the 14 open problems posted by Hiriart-Urruty [6] nine years ago: "A collection of m symmetric (n, n) matrices {A 1 , A 2 , . . . , A m } is said to be simultaneously diagonalizable via congruence if there exists a nonsingular matrix P such that each of the P T A i P is diagonal." Two sufficient conditions for SD of two matrices A and B have been developed in [6,13,14,17]: i) There exist µ 1 and µ 2 ∈ ℜ such that µ 1 A+ µ 2 B ≻ 0, which is termed as regular case in [13,14,17]; and ii) When n ≥ 3, ( Ax, x = 0 and Bx, x = 0) implies (x = 0). Lancaster and Rodman [10] offer another sufficient condition for SD of two matrices A and B: There exist α and β such that C := αA + βB 0 and Ker(C) ⊆ Ker(A) ∩ Ker(B). Several other SD sufficient conditions of two matrices are presented in Theorem 4.5.15 of [7]. A necessary and sufficient condition for SD of two matrices has been derived by Uhlig in [20,21] if at least one of the two matrices is nonsingular. A sufficient SD condition is well known for multiple matrices: If m matrices commute with each other, then they are SD, see [7] (Problems 22 and 23, Page 243). The equivalence between the attainability of SD and a hidden convexity

In this paper, we provide the first provable linear-time (in term of the number of non-zero entries of the input) algorithm for approximately solving the generalized trust region subproblem (GTRS) of minimizing a quadratic function over a quadratic constraint under some regularity condition. Our algorithm is motivated by and extends a recent linear-time algorithm for the trust region subproblem by Hazan and Koren [Math. Program., 2016, 158(1-2): 363-381]. However, due to the non-convexity and non-compactness of the feasible region, such an extension is nontrivial. Our main contribution is to demonstrate that under some regularity condition, the optimal solution is in a compact and convex set and lower and upper bounds of the optimal value can be computed in linear time. Using these properties, we develop a linear-time algorithm for the GTRS.

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